Development of meshfree strong form methods

DEVELOPMENT OF MESHFREE STRONG-FORM METHODS KEE BUCK TONG, BERNARD (B. Eng. (Hons.), NUS) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MACHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2007 Table of Content Acknowledgements I would like to express my deepest gratitude to my supervisor, Prof. Liu Guirong, for his dedicated support, guidance and continuous encouragement during my Ph.D. study. To me, Prof. Liu is also a kind mentor who inspires me not only in my research work but also in many aspects of my life. I would also like to extend a great thank to my co-supervisor, Dr. Lu Chun, for his valuable advices in many aspects of my research work. I would also like to give many thanks to my fellow colleagues and friends in Center for ACES, Dr. Gu Yuan Tong, Dr. Liu Xin, Dr. Dai Keyang, Dr. Zhang Guiyong, Dr. Zhao Xin, Dr. Deng Bin, Mr. Li Zirui, Mr. Zhang Jian, Mr. Khin Zaw, Mr. Song Chengxiang, Ms. Chen Yuan, Mr. Phuong, Mr. Trung, Mr. Chou Cheng-En, Mr. George Xu. The constructive suggestions, professional opinions, interactive discussions among our group definitely help to improve the quality of my research work. And most importantly, these guys have made my life in Center for ACES a joyful one. I am also indebted to many of my close friends, friends from JBKakis, Man Woei, Kuang Hoe, You Mao, who continuously encourage and motivate me to keep up the good job. Without this valuable friendship and love, my life is not going to be stimulating, interesting and enjoyable. Great appreciation is extended to my dearest family members, my parents, my sisters, Susanna, Kathy and Karen for their strong support and cares. Not to mention, I own very much to my lovely fiancée, Michelle Ding, who is always giving me strong i Table of Content support, great tolerance, cares and understanding. It is impossible for me to complete this work without her love. This piece of work is also a present for our wedding. Lastly, I appreciate the National University of Singapore for granting me research scholarship whick makes my Ph.D. study possible. Thanks A*STAR for the pre-graduate scholarship which supports me during the last year of my undergraduate study. Many thanks are conveyed to Mechanical department and Center for ACES for their material support to every aspect of this work. ii Table of Content Table of contents Acknowledgements i Table of contents . iii Summary ix Nomenclature xiii List of Figures .xvi List of Tables . xxviii Chapter 1.1 Introduction .1 Background 1.1.1 Motivation of Meshfree Methods . 1.1.2 Features of Meshfree Methods . 1.2 Literature review 1.2.1 Classification of Meshfree Methods 1.2.2 Meshfree Weak-form Methods . 1.2.3 Meshfree Strong-form Methods . 1.2.4 Meshfree Weak-Strong Form Methods 1.3 Motivation of the Thesis 1.4 Objectives of the Thesis . 11 1.5 Organization of the Thesis 13 Chapter Function Approximations .16 2.1 Introduction 16 2.2 Smooth Particle Hydrodynamics (SPH) Approximation .17 iii Table of Content 2.3 Reproducing Kernel Particle Method (RKPM) Approximation .18 2.4 Moving Least-Squares (MLS) Approximation 19 2.5 Polynomial Point Interpolation Method (PPIM) Approximation .21 2.5.1 Formulation of Polynomial Point Interpolation Method 22 2.5.2 Properties of PPIM Shape Function . 24 2.5.3 Techniques to Overcome Singularity in Moment Matrix . 26 2.6 Radial Point Interpolation Method (RPIM) Approximation .27 2.6.1 Formulation of Radial Point Interpolation Method 28 2.6.2 Property of RPIM Shape Function . 30 2.6.3 Radial Basis Functions . 32 2.6.4 Implementation Issues of RPIM Approximation . 33 2.6.5 Comparison between RPIM and PPIM Shape Functions . 34 Chapter Adaptivity .39 3.1 Introduction 39 3.2 Definition of Errors 40 3.3 Error Estimators 42 3.3.1 Interpolation Variance Based Error Estimator 43 3.3.1.1 Formulation of Interpolation Variance Based Error Estimator 43 3.3.1.2 Remarks 44 3.3.2 Residual Based Error Estimator . 45 3.3.2.1 Formulation of Residual Based Error Estimator . 46 3.3.2.2 Numerical Examples: 47 3.3.2.3 Remarks 55 3.4 Adaptive Strategy .57 3.4.1 Local Refinement Criterion 57 3.4.2 Stopping Criterion 58 3.5 Refinement Procedure .58 iv Table of Content 3.5.1 Refinement Procedure for Interpolation Variance based Error Estimator 59 3.5.2 Refinement Procedure for Residual based Error Estimator . 59 Chapter Radial Point Collocation Method (RPCM) .73 4.1 Introduction 73 4.2 Formulation of RPCM .74 4.3 Issues in RPCM 76 4.4 Numerical Examples: .79 4.4.1 Example 1: One Dimensional Poisson Problem 79 4.4.2 Example 2: Two dimensional Poisson Problem with Dirichlet Boundary Conditions . 81 4.4.3 Example 3: Standard and Higher Order Patch Tests 82 4.4.4 Example 4: Elastostatics Problem with Neumann Boundary Conditions 84 4.5 Remarks: .86 Chapter A Stabilized Least-Squares Radial Point Collocation Method (LS-RPCM) .94 5.1 Introduction 94 5.2 Stabilized Least-squares Procedure .95 5.3 Numerical Examples 100 5.3.1 Example 1: A Cantilever Beam Subjected to a Parabolic Shear Stress at the Right End 100 5.3.2 Example 2: Poisson Problem with Neumann Boundary Conditions 103 5.3.3 Example 3: Infinite Plate with Hole Subjected to an Uniaxial Traction in the Horizontal Direction 105 5.3.4 Example 4: A L-shaped Plate Subjected to a Unit Tensile Traction in the Horizontal Direction 106 5.4 Remarks 107 v Table of Content Chapter A Least-Square Radial Point Collocation Method (LS-RPCM) with Special Treatment for Boundaries 119 6.1 Introduction 119 6.2 Least-square Procedure with Special Treatment for Boundaries 120 6.3 Numerical Examples 123 6.3.1 Example 1: Infinite Plate with Hole Subjected to a Uniaxial Traction in the Horizontal Direction 124 6.3.2 Example 2: Cantilever Beam Subjected to a Parabolic Shear Traction at the End 125 6.3.3 Example 3: Poisson Problem with Smooth Solution . 127 6.3.4 Example 4: A Thick Wall Cylinder Subjected an Internal Pressure . 128 6.3.5 Example 5: A Reservoir Full Filled with Water . 130 6.4 Remarks 131 Chapter A Regularized Least-Square Radial Point Collocation Method (RLS-RPCM) 151 7.1 Introduction 151 7.2 Regularization Procedure 152 7.2.1 Regularization Equations . 152 7.2.2 Regularization Least-square Formulation 154 7.2.3 Determination of Regularization Factor . 155 7.3 Numerical Examples 156 7.3.1 Example 1: Cantilever Beam . 157 7.3.2 Example 2: Hollow Cylinder with Internal Pressure 159 7.3.3 Example 3: Bridge with Uniform Loading on the Top . 160 7.3.4 Example 4: Poisson Problem with High Gradient Solution . 161 7.3.5 Example 5: Poisson Problem with Multiple Peaks Solution 163 vi Table of Content 7.4 Remarks 165 Chapter A Subdomain Method Based on Local Radial Basis Functions .184 8.1 Introduction 184 8.2 Formulation of Subdomain Method .186 8.3 Numerical Examples 192 8.3.1 Example 1: Standard and Higher order Patch Tests . 193 8.3.2 Example 2: Connecting Rod Subjected to Internal Pressure 193 8.3.3 Example 3: A Cantilever Beam Subjected to a Parabolic Shear at End . 194 8.3.4 Example 4: Adaptive Analysis of Elastostatics Problem 195 8.3.5 Example 5: Adaptive Analysis of Short Beam Subjected to Uniform Loading on the Top Edge 196 8.3.6 Example 6: Adaptive Analysis of Bridge Subjected to Uniform Loading on the Top Edge .198 8.3.7 8.4 Example 7: Adaptive Analysis of Crack Problem 199 Remarks 200 Chapter Effects of the Number of Local Nodes for Meshfree Methods Based on Local Radial Basis Functions 220 9.1 Introduction 220 9.2 Nodal Selection .222 9.3 Concept of Layer 224 9.4 Numerical Examples 225 9.3.1 Examples 1: Curve Fitting . 225 9.3.2 Examples 2: LC-RPIM (Weak-form Method) for Elastostatics Problem . 227 9.3.3 Examples 3: RPCM (Strong-form Method) for Torsion Problem 228 9.3.4 Examples 4: RLS-RPCM (Strong-form) for Elastostatics Problem . 231 vii Table of Content 9.3.5 9.5 Examples 5: Adaptive RPCM for Dirichlet Problem . 232 Remarks 234 Chapter 10 Conclusion and Future Work .255 10.1 Conclusion Remarks 255 10.2 Recommendation for future work 259 References 261 Publications Arising From Thesis .270 viii Summary Summary Meshfree method is a new promising numerical method after the finite element method (FEM) has been dominant in computational mechanics for several decades. The feature of mesh free has drawn a lot of attention from mathematicians and researchers. Development of meshfree method has achieved remarkable success in recent years. Among the meshfree methods, the progress of the development of meshfree strong-form method is still very sluggish. As compared to meshfree weak-form method, the relevant research works dedicated to meshfree strong-form method are still not abundantly available in the literature. Nonetheless, strong-form meshfree method possesses many attractive and distinguished features that facilitate the implementation of the adaptive analysis. In this study, the two primary objectives are: (1) To provide remedies to stabilize the solution of strong-form meshfree method (2) To extend strong-form meshfree method to adaptive analysis Radial point collocation method (RPCM) is a strong-form meshfree method studied in this work. Instability is a fatal shortcoming that prohibits RPCM from being used in adaptive analysis. The first contribution of this thesis is to propose several techniques that can be employed to stabilize the solution of RPCM before it can be used in adaptive analysis. Stabilized least-squares RPCM (LS-RPCM) is the first proposed meshfree strong-form method that uses stabilization least-squares technique to restore the stability of RPCM solution. In the stabilization procedure, additional governing ix Chapter 10 Conclusion and Future Work small number is recommended and α = 0.05 is used in this work. The advantages of strong-form meshfree method have been evidently shown in the large number of numerical examples. (2) As Neumann boundary condition is blamed for the cause of instability, a least-square procedure with special treatment for boundaries is proposed to stabilize the solution of RPCM [68]. In the second stabilization procedure, more collocation points (not nodes) are introduced on the boundaries to reduce the ‘strong’ requirement of satisfying the governing equation and boundary conditions, and hence a relaxation effect is provided. The well-known least-square technique is applied to solve for the over-determined algebraic equations. Although the boundary conditions and the governing equation are not fully satisfied on nodes, the solution of the LS-RPCM is stable and in good accuracy. As stability is restored, the LS-RPCM is also extended to adaptive analysis. Examples illustrated in this work have clearly shown that good result can be obtained in the adaptive LS-RPCM. (3) Regularization technique that commonly be used for solving ill-posed inversed problem is used to stabilize the solution of RPCM in this work [67]. The regularization procedure that adopts Tikhonov regularization technique is introduced in the regularization least-squares RPCM (RLS-RPCM). As special regularization equations and regularization points are suggested, no regularization factor has to be determined in the 256 Chapter 10 Conclusion and Future Work RLS-RPCM. After stable and accurate solution is obtained through the proposed regularization procedure, RLS-RPCM has also successfully applied in adaptive analysis. A vast number of examples have shown adaptive RLS-RPCM is efficient and good numerical performance is demonstrated. (4) In addition to the strong-form meshfree methods, a classical subdomain method integrated with RPIM shape functions is proposed [71]. By applying the meshfree techniques in classical subdomain method, the present method has demonstrated good stability and accuracy. Numerous numerical examples have shown the subdomain method can be easily extended to adaptive analysis and good results can be obtained. Subdomain method has also been shown robust, stable and accurate as compared to stabilized RPCM. One of the main reasons may due to the formulation procedure that integrated Neumann boundary condition naturally. The residual in the subdomain is also kept to minimum. For strong form method, the problem domain is only represented by nodes and residual is kept to minimum at nodes only. Nevertheless, strong form method has also possesses several attractive features such as feature of free from domain discretization and integration. (5) Before all the meshfree strong-form method can be extended to adaptive analysis, a robust and effective error estimator that is customized for the strong-form method has to be developed. This is one of the most 257 Chapter 10 Conclusion and Future Work challenging works in this thesis as most of the well established error estimator is only applicable for weak-form and based on mesh. The residual based error estimator proposed in this work is an excellent error estimator [69]. It has been shown effective and robust in the numerous numerical examples of various meshfree strong-form methods. In addition, this versatile error estimator has been successfully implemented in the adaptive FEM and adaptive subdomain method as well. As compared to the conventional error estimators used in the adaptive FEM, the present error estimator has also exhibited great advantages in terms of computational cost and efficiency. (6) RPIM approximation is used in all the strong-form methods and the proposed subdomain method to construct their shape functions. In RPIM approximation based on the local RBFs, RBFs has significantly influences to the solutions. Hence, a thorough study of the local RBFs is very important. Although the shape parameters of the local RBFs have been comprehensively studied in literature, the effects of the number of local nodes for meshfree methods are not intensively investigated. In this work, an insightful and comprehensive study on the effects of the number of local nodes for meshfree methods based on local RBFs is provided [70]. The local RBFs not only reduce the computational cost drastically, but also provide a more stable coefficient matrix, especially in the strong-form meshfree methods. Instead of suggesting how many local 258 Chapter 10 Conclusion and Future Work nodes to be used, a concept of ‘layer’ is introduced to facilitate the nodal selection in the function approximation. 10.2 Recommendation for future work Based on the presented work in this thesis, the following recommendations are given for future work: (1) All the examples given in this thesis are one or two dimensional problems. It is possible to extend the presented strong-form meshfree methods to solve for three dimensional problems. The proposed stabilization techniques are still applicable; however, some modifications are required. As the formulation of strong-form method is simple and straightforward, low computational cost is expected in the three dimensional cases. (2) It is also possible to extend my adaptive works to non-linear and dynamic problems. However, a more complicated adaptive strategy has to be devised. In the dynamics problems, as the field function is varying at different time, coarsening procedure is required to remove the unnecessary nodes in the problem domain. (3) Since knowledge of meshfree techniques has been gained through the experience in the past, it is possible to apply those techniques to the classical method, e.g., FEM, subdomain method. Besides the RPIM approximation, it is also possible to use PPIM approximation in the subdomain formulation with some special treatments on the boundaries. 259 Chapter 10 Conclusion and Future Work (4) Although several stabilization techniques are suggested in the presented work, I believe that these proposed stabilization procedures are not the only possible approaches to provide a stable strong-form solution. As development of strong-form meshfree methods is still at the developing stage, more stabilization procedures or novel formulations may be proposed in the future. A stable strong-form solution is still very much desired and in great demand. 260 References References Ainsworth M and Oden JT (1993) A unified approach to a posteriori error estimation using element residual methods, Numer. Math, 65: 23-50. 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Methods Engrg., 24: 337-357. 102 Zienkiewicz OC and Zhu JZ (1992) The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique, Int. J. Numer. Methods Engrg., 33: 1331-1364. 103 Zienkiewicz OC and Zhu JZ (1992) The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity, Int. J. Numer. Methods Engrg., 33: 1365-1382. 104 Zienkiewicz OC and Zhu JZ (1992) The superconvergent patch recovery (SPR) and adaptive finite element refinement, Comput. Methods Appl. Mech. Engrg. 101: 207-224. 269 Publications arising from thesis Publications Arising From Thesis Conference Papers: [C1] Liu GR, Kee BBT (2005) An adaptive meshfree least-square method (keynote), Computational & Experimental Mechanics 2005, UKM, Bangi, Malaysia, p1-9, 2005 May. [C2] Liu GR, Kee BBT (2005) An adaptive meshfree strong-form method based on regularized least-squares procedure, International Conference on Computational & Experimental Engineering and Sciences 2005, Chennai, India, p207-212, 2005 Dec. [C3] Liu GR, Kee BBT (2006) A regularized strong-form meshfree method for adaptive analysis (keynote), III European Conference on Computational Mechanics (ECCM) Solids, Structures and Coupled Problems in Engineering, Lisbon, Portugal, 2006 July. [C4] Liu GR, Kee BBT, Zhong ZH, Li GY, Han X (2006) Adaptive meshfree methods using local nodes and radial basis functions (semi-plenary lecture), Computational Methods in Engineering and Science EPMESC, Sanya, Hainan, China, Aug 2006. Journal Papers: [J1] Liu GR, Kee BBT, Lu C (2006) A stabilized least-squares radial point collocation method (LS-RPCM) for adaptive analysis, Comput. Methods Appl. Mech. Engrg., 195: 4843-4861 [J2] Kee BBT, Liu GR, Lu C (2007) A regularized least-square radial point collocation method (RLS-RPCM) for adaptive analysis, Comp. Mech., (Available Online). [J3] Kee BBT, Liu GR, Lu C (2007) A Least-square Radial Point Collocation Method for 270 Publications arising from thesis Adaptive Analysis in Linear Elasticity, Engineering Analysis with Boundary Elements, (Accepted) [J4] Kee BBT, Liu GR, Lu C (2007) A Residual Based Error Estimator Using Radial Basis Functions, Finite Elements in Analysis and Design, (Submitted) [J5] Kee BBT, Liu GR, Song CX, Zhang J, Zhang GY (2007) A Study on the Effect of the Number of Local Nodes for Meshfree Methods Based on Radial Basis Functions, Computer modeling in Engineering & Science (Summitted) [J6] Bernard B. T. Kee, G. R. Liu, C. Lu (2007) An Adaptive Subdomain Method Based Local Radial Basis Functions, (In writing process) 271 [...]... the meshfree techniques has demonstrated great numerical performance in term of accuracy and stability x Summary The second significant contribution of this work is the development of an error estimator for strong- form meshfree method Most of the existing error estimators for adaptive meshfree method are an extension of the conventional error estimators for FEM which is formulated in term of weak -form. .. Liszka and Orkizs [37,38] etc The purpose of this section is just to provide a brief history of meshfree methods More comprehensive overview of the development of meshfree methods is abundantly available in literature [8,40,46] 1.2.1 Classification of Meshfree Methods As meshfree method is developing progressively, it is very important to classify the meshfree methods into different categories for better... of classification that categorizes the meshfree methods according to the domain representation This type of classification categorizes the meshfree method into two categories: domain-type and boundary-type of meshfree methods In domain-type of meshfree methods, both problem domain and boundary are will represented by field nodes Examples of this type of meshfree methods includes element-free Galerkin... needed in some meshfree methods, the implementation of adaptive analysis and boundary moving problems should be done with ease (4) Meshfree methods should provide a better accuracy for the solution of the derivative of the primary field function such as stress (5) Meshfree methods should be able to provide solution with higher accuracy for high deformation problem The accuracy of the meshfree method’s... of the use of mesh As an error estimator that is feasible for strong- form meshfree method is available and stability of the RPCM solution is restored, all the presented meshfree strong- form methods and subdomain method have been successfully extended to adaptive analysis All the presented adaptive meshfree methods have been shown to be very simple and easy to implement due to the features of mesh free... numerical method, meshfree method, which is formulated without using the mesh, is therefore in great demand 1.1.2 Features of Meshfree Methods The motivation of the meshfree methods has been clearly stated in the last section Furthermore, close examination has revealed the difficulties caused by the use of mesh in the FEM To get rid of the mesh, a new class of the numerical method, meshfree method,... (FVM), are no longer suited well The demand of new class of numerical method that is formulated without the reliance of mesh or grid becomes more significant This motivation drives the leap of the meshfree methods in the last three decades Meshfree method has become one of the hottest research topics in the computational mechanics community and many meshfree methods have been well established and discussed... contribution to extend the application of SPH method [21,76-77] A comprehensive discussion of the recent research works of SPH method can be found in Ref [54] Besides the SPH method, the collocation method is another well-known meshfree method which has great influence to the development of meshfree methods As early as 80s, to get rid of the regular grids in the formulation of finite difference method (FDM),... boundary-type of meshfree methods, for example, boundary node method (BNM) [78], boundary point interpolation method (BPIM) [25], boundary radial point interpolation method (BRPIM) [26] In this thesis, meshfree methods are classified according to the formulation procedure is adopted They can be largely categorized into three different categories, namely meshfree weak -form method, meshfree strong- form method... matrix of the LC-RPIM using different number of local nodes Figure 9.12 Distribution of a set of 100 randomly scattered nodes in a square domain 245 245 Figure 9.13 (a) Dimension of a triangular cross section bar, (b) a model of the bar with 120 field nodes 246 xxv List of Figures Figure 9.14 Error norm of the RPCM solution with different number of local nodes 246 Figure 9.15 Computational time of the . Features of Meshfree Methods 3 1.2 Literature review 5 1.2.1 Classification of Meshfree Methods 6 1.2.2 Meshfree Weak -form Methods 8 1.2.3 Meshfree Strong- form Methods 8 1.2.4 Meshfree Weak -Strong. stabilize the solution of strong- form meshfree method (2) To extend strong- form meshfree method to adaptive analysis Radial point collocation method (RPCM) is a strong- form meshfree method studied. of the development of meshfree strong- form method is still very sluggish. As compared to meshfree weak -form method, the relevant research works dedicated to meshfree strong- form method are still